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प्रश्न
In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?
विकल्प
(a) 35
(b) 48
(c) 60
(d) 22
(e) 30
उत्तर
(c) 60
Let M, P and C denote the sets of students who have opted for mathematics, physics, and chemistry, respectively.
Here,
\[n\left( M \right)\] = 100,
\[n\left( P \right)\]=70 and \[n(C)\] = 40
Now,\[n\left( M \cap P \right) = 30, n\left( M \cap C \right) = 28, n\left( P \cap C \right) = 23 and n\left( M \cap P \cap C \right) = 18\]Number of students who opted for only mathematics:
\[n\left( M \cap P' \cap C' \right) = n\left\{ M \cap \left( P \cup C \right)' \right\}\]
\[ = n\left( M \right) - n\left\{ M \cap \left( P \cup C \right) \right\}\]
\[ = n\left( M \right) - n\left\{ \left( M \cap P \right) \cup \left( M \cap C \right) \right\}\]
\[ = n\left( M \right) - \left\{ n\left( M \cap P \right) + n\left( M \cap C \right) - n\left( M \cap P \cap C \right) \right\}\]
\[ = 100 - \left( 30 + 28 - 18 \right)\]
\[ = 60\]
Therefore, the number of students who opted for mathematics alone is 60
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